arXiv subjects

Shangshuai Li

Publications and source records attributed to Shangshuai Li.

8 recordsLinked to original sources

arXiv2026

Sato-theoretic construction of the anti-self-dual Yang-Mills hierarchy and the Ward conjecture

We develop the Sato-theoretic dressing framework for the anti-self-dual Yang-Mills (ASDYM) hierarchy based on a normalized Riemann-Hilbert decomposition. A four-sector expansion of the generating function yields a bi-infinite matrix array of relative coordinates, extending the affine coordinates on the big cell of the Sato Grassmannian. We derive the Sato-Wilson equations, continuous coordinate flows, and their discrete analogues. Several classical integrable hierarchies are recovered under dimensional reduction constraints, with their nonlinear variables identified as specific relative coordinates.

nlin.SI↗

arXiv2026

A unified approach to the AKNS, DNLS, KP and mKP hierarchies in the anti-self-dual Yang-Mills reduction

We show a unified approach to the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and the unreduced derivative nonlinear Schrödinger (DNLS) hierarchies (including the Kaup-Newell, Chen-Lee-Liu, Gerdjikov-Ivanov and a generalized DNLS), together with their multi-component extensions, in the framework of the anti-self-dual Yang-Mills (ASDYM) reduction. By restricting the gauge group to GL(2), the Kadomtsev-Petviashvili (KP) and modified KP (mKP) hierarchies are formulated in the ASDYM reduction via squared eigenfunction symmetry constraints. In this case, the bilinearization of the generalized DNLS equations can also be understood through this reduction. Finally, Gram-type exact solutions for the relevant equations are presented in terms of quasi-determinants.

nlin.SI↗

arXiv2025

Soliton Resonances in Four Dimensional Wess-Zumino-Witten Model

We present two kinds of resonance soliton solutions on the Ultrahyperbolic space $\mathbb{U}$ for the G=U(2) Yang equation, which is equivalent to the anti-self-dual Yang-Mills (ASDYM) equation. We reveal and illustrate the solitonic behaviors in the four-dimensional Wess-Zumino-Witten (WZW$_4$) model through the sigma model action densities. The Yang equation is the equation of motion of the WZW$_4$ model. In the case of $\mathbb{U}$, the WZW$_4$ model describes a string field theory action of open N=2 string theories. Hence, our solutions on $\mathbb{U}$ suggest the existence of the corresponding classical objects in the N=2 string theories. Our solutions include multiple-pole solutions and V-shape soliton solutions. The V-shape solitons suggest annihilation and creation processes of two solitons and would be building blocks to classify the ASDYM solitons, like the role of Y-shape solitons in classification of the KP (line) solitons. We also clarify the relationship between the Cauchy matrix approach and the binary Darboux transformation in terms of quasideterminants. Our formalism can start with a simpler input data for the soliton solutions and hence might give a suitable framework for the classification of the ASDYM solitons.

hep-th↗

arXiv2024

From the self-dual Yang-Mills equation to the Fokas-Lenells equation

A reduction from the self-dual Yang-Mills (SDYM) equation to the unreduced Fokas-Lenells (FL) system is described in this paper. It has been known that the SDYM equation can be formulated from the Cauchy matrix schemes of the matrix Kadomtsev-Petviashvili (KP) hierarchy and the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy. We show that the reduction can be realized in these two Cauchy matrix schemes, respectively. Each scheme allows us to construct solutions for the unreduced FL system. We prove that these solutions obtained from different schemes are equivalent under certain reflection transformation of coordinates. Using conjugate reduction we obtain solutions of the FL equation. The paper adds an important example to Ward's conjecture on the reductions of the SDYM equation. It also indicates the Cauchy matrix structures of the Kaup-Newell hierarchy.

nlin.SI↗

arXiv2024

Direct linearization of the SU(2) anti-self-dual Yang-Mills equation in various spaces

The paper establishes a direct linearization scheme for the SU(2) anti-self-dual Yang-Mills (ASDYM) equation.The scheme starts from a set of linear integral equations with general measures and plane wave factors. After introducing infinite-dimensional matrices as master functions, we are able to investigate evolution relations and recurrence relations of these functions, which lead us to the unreduced ASDYM equation. It is then reduced to the ASDYM equation in the Euclidean space and two ultrahyperbolic spaces by reductions to meet the reality conditions and gauge conditions, respectively. Special solutions can be obtained by choosing suitable measures.

nlin.SI↗

arXiv2023

Solutions to the SU($\mathcal{N}$) self-dual Yang-Mills equation

In this paper we aim to derive solutions for the SU($\mathcal{N}$) self-dual Yang-Mills (SDYM) equation with arbitrary $\mathcal{N}$. A set of noncommutative relations are introduced to construct a matrix equation that can be reduced to the SDYM equation. It is shown that these relations can be generated from two different Sylvester equations, which correspond to the two Cauchy matrix schemes for the (matrix) Kadomtsev-Petviashvili hierarchy and the (matrix) Ablowitz-Kaup-Newell-Segur hierarchy, respectively. In each Cauchy matrix scheme we investigate the possible reductions that can lead to the SU$(\mathcal{N})$ SDYM equation and also analyze the physical significance of some solutions, i.e. being Hermitian, positive-definite and of determinant being one.

nlin.SI↗

arXiv2021

Cauchy matrix approach to the SU(2) self-dual Yang--Mills equation

The Cauchy matrix approach is developed to solve the SU(2) self-dual Yang--Mills equation. Starting from a Sylvester matrix equation coupled with certain dispersion relation for infinite coordinates, the self-dual Yang--Mills equation under Yang's formulation is constructed. By imposing further constraints on complex independent variables, a broad class of explicit solutions are obtained.

nlin.SI↗

arXiv2020

Symmetric discrete AKP and BKP equations

We show that when KP (Kadomtsev-Petviashvili) $τ$ functions allow special symmetries, the discrete BKP equation can be expressed as a linear combination of the discrete AKP equation and its reflected symmetric forms. Thus the discrete AKP and BKP equations can share the same $τ$ functions with these symmetries. Such a connection is extended to 4 dimensional (i.e. higher order) discrete AKP and BKP equations in the corresponding discrete hierarchies. Various explicit forms of such $τ$ functions, including Hirota's form, Gramian, Casoratian and polynomial, are given. Symmetric $τ$ functions of Cauchy matrix form that are composed of Weierstrass $σ$ functions are investigated. As a result we obtain a discrete BKP equation with elliptic coefficients.

nlin.SI↗